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Parallel and Intersecting Lines - Across the Line

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Intersecting lines are lines that cross each other at exactly one point, known as the point of intersection. At this point, they form four angles. The pairs of angles directly opposite each other are called vertically opposite angles and are always equal: ∠1=∠3\angle 1 = \angle 3 and ∠2=∠4\angle 2 = \angle 4.

Two intersecting lines forming vertically opposite angles.
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Parallel lines are lines in a plane that never meet, no matter how far they are extended. The perpendicular distance between them remains constant. Symbolically, we write l∥ml \parallel m.

Two parallel lines l and m.
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A transversal is a line that intersects two or more lines at distinct points. When a transversal intersects two parallel lines, several angle relationships are formed: Corresponding angles are equal, Alternate Interior angles are equal, and Co-interior angles are supplementary (180∘180^\circ).

A transversal line t intersecting two parallel lines l and m showing alternate interior angles a and b.
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Angles on a straight line always add up to 180∘180^\circ. This is known as a linear pair when two angles are adjacent and their non-common sides form a line.

📐Formulae

If l∥m, then ∠Corresponding1=∠Corresponding2\text{If } l \parallel m, \text{ then } \angle \text{Corresponding}_1 = \angle \text{Corresponding}_2

∠Alternate Interior1=∠Alternate Interior2\angle \text{Alternate Interior}_1 = \angle \text{Alternate Interior}_2

∠Co-interior1+∠Co-interior2=180∘\angle \text{Co-interior}_1 + \angle \text{Co-interior}_2 = 180^\circ

Sum of angles in a linear pair=180∘\text{Sum of angles in a linear pair} = 180^\circ

💡Examples

Problem 1:

In the given figure, line ll is parallel to line mm (l∥ml \parallel m) and a transversal pp cuts them. If one of the interior angles is 125∘125^\circ, find the measure of its co-interior angle xx.

Solution:

Since l∥ml \parallel m, the sum of co-interior angles is 180∘180^\circ. 125∘+x=180∘125^\circ + x = 180^\circ x=180∘−125∘x = 180^\circ - 125^\circ 180−12555\begin{array}{r} 180 \\ - 125 \\ \hline 55 \end{array} x=55∘x = 55^\circ

Explanation:

When two parallel lines are intersected by a transversal, the interior angles on the same side of the transversal are supplementary, meaning their sum is 180∘180^\circ.

Problem 2:

Two lines intersect at a point. If one angle is 45∘45^\circ, find the measures of the other three angles.

Solution:

Let the given angle be ∠1=45∘\angle 1 = 45^\circ.

  1. ∠3\angle 3 is vertically opposite to ∠1\angle 1, so ∠3=45∘\angle 3 = 45^\circ.
  2. ∠2\angle 2 forms a linear pair with ∠1\angle 1, so ∠1+∠2=180∘\angle 1 + \angle 2 = 180^\circ. ∠2=180∘−45∘=135∘\angle 2 = 180^\circ - 45^\circ = 135^\circ
  3. ∠4\angle 4 is vertically opposite to ∠2\angle 2, so ∠4=135∘\angle 4 = 135^\circ.

Explanation:

Vertically opposite angles are equal, and angles on a straight line (linear pair) add up to 180∘180^\circ.

Problem 3:

If a transversal intersects two lines such that a pair of alternate interior angles are 3x+10∘3x + 10^\circ and 2x+30∘2x + 30^\circ, find the value of xx for which the lines are parallel.

Solution:

For the lines to be parallel, the alternate interior angles must be equal. 3x+10=2x+303x + 10 = 2x + 30 3x−2x=30−103x - 2x = 30 - 10 x=20x = 20

Explanation:

The property of parallel lines states that alternate interior angles are equal. By setting the two expressions equal to each other, we solve for the variable xx.

Problem 4:

In the following figure, line pp is parallel to line qq (p∥qp \parallel q). If the measure of ∠1=75∘\angle 1 = 75^\circ, find the measure of ∠2\angle 2.

Parallel lines p and q with a transversal showing co-interior angles 1 and 2.

Solution:

Given p∥qp \parallel q and ∠1=75∘\angle 1 = 75^\circ. From the figure, ∠1\angle 1 and ∠2\angle 2 are interior angles on the same side of the transversal (co-interior angles). We know that co-interior angles are supplementary: ∠1+∠2=180∘\angle 1 + \angle 2 = 180^\circ 75∘+∠2=180∘75^\circ + \angle 2 = 180^\circ ∠2=180∘−75∘\angle 2 = 180^\circ - 75^\circ ∠2=105∘\angle 2 = 105^\circ

Explanation:

Because the lines are parallel, the angles located between the lines on the same side of the intersecting line (transversal) must sum to 180 degrees.

Problem 5:

In the given figure, line ABAB and CDCD intersect at point OO. If ∠AOC=2x−10∘\angle AOC = 2x - 10^\circ and ∠BOD=70∘\angle BOD = 70^\circ, find the value of xx.

Intersecting lines AB and CD at point O showing vertically opposite angles.

Solution:

Lines ABAB and CDCD intersect at OO. Therefore, ∠AOC\angle AOC and ∠BOD\angle BOD are vertically opposite angles. Since vertically opposite angles are equal: ∠AOC=∠BOD\angle AOC = \angle BOD 2x−10∘=70∘2x - 10^\circ = 70^\circ 2x=70∘+10∘2x = 70^\circ + 10^\circ 2x=80∘2x = 80^\circ x=80∘2x = \frac{80^\circ}{2} x=40∘x = 40^\circ

Explanation:

When two straight lines cross, the angles opposite the vertex are equal. By setting the expression for one angle equal to the value of its opposite angle, we can solve for the unknown variable.